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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arymaptfv | Structured version Visualization version GIF version | ||
| Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.) | 
| Ref | Expression | 
|---|---|
| 1arymaptfv.h | ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) | 
| Ref | Expression | 
|---|---|
| 1arymaptfv | ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq1 6904 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{〈0, 𝑥〉}) = (𝐹‘{〈0, 𝑥〉})) | |
| 2 | 1 | mpteq2dv 5243 | . 2 ⊢ (ℎ = 𝐹 → (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉})) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) | 
| 3 | 1arymaptfv.h | . 2 ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉}))) | |
| 4 | eqid 2736 | . . . . 5 ⊢ (0..^1) = (0..^1) | |
| 5 | 4 | naryrcl 48557 | . . . 4 ⊢ (ℎ ∈ (1-aryF 𝑋) → (1 ∈ ℕ0 ∧ 𝑋 ∈ V)) | 
| 6 | 5 | simprd 495 | . . 3 ⊢ (ℎ ∈ (1-aryF 𝑋) → 𝑋 ∈ V) | 
| 7 | 6 | mptexd 7245 | . 2 ⊢ (ℎ ∈ (1-aryF 𝑋) → (𝑥 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉})) ∈ V) | 
| 8 | 2, 3, 7 | fvmpt3 7019 | 1 ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 〈cop 4631 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 ℕ0cn0 12528 ..^cfzo 13695 -aryF cnaryf 48552 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-naryf 48553 | 
| This theorem is referenced by: 1arymaptf1 48568 | 
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